Use DeMoivre's Theorem To Calculate The Following Expression. Write The Exact Answer In The Form Found Using Euler's Formula, |z|eiθ . Do Not Round. Make Sure That The Argument Of Your Answer Lies In The Interval [0,2π) . [2(cos(π3)+isin(π3))]4
Introduction
DeMoivre's Theorem is a powerful tool in complex analysis that allows us to raise complex numbers to powers. It states that for any complex number z = r(cosθ + isinθ) and any integer n, we have:
z^n = r^n(cos(nθ) + isin(nθ))
In this article, we will use DeMoivre's Theorem to calculate the expression [2(cos(π/3) + isin(π/3))]^4 and write the exact answer in the form found using Euler's Formula, |z|eiθ.
Euler's Formula
Euler's Formula is a fundamental result in complex analysis that states that for any complex number z = r(cosθ + isinθ), we have:
z = re^(iθ)
where e is the base of the natural logarithm and i is the imaginary unit.
DeMoivre's Theorem
DeMoivre's Theorem is a direct consequence of Euler's Formula. It states that for any complex number z = r(cosθ + isinθ) and any integer n, we have:
z^n = r^n(cos(nθ) + isin(nθ))
This theorem allows us to raise complex numbers to powers in a straightforward and efficient way.
Calculating the Expression
To calculate the expression [2(cos(π/3) + isin(π/3))]^4, we can use DeMoivre's Theorem directly. We have:
r = 2 θ = π/3 n = 4
Using DeMoivre's Theorem, we get:
[2(cos(π/3) + isin(π/3))]^4 = 2^4(cos(4(π/3)) + isin(4(π/3)))
= 16(cos(4π/3) + isin(4π/3))
= 16(-1/2 + i√3/2)
= -8 + 8i√3
However, we want to write the answer in the form found using Euler's Formula, |z|eiθ. To do this, we need to find the argument of the complex number -8 + 8i√3.
Finding the Argument
The argument of a complex number z = x + iy is the angle θ between the positive x-axis and the line segment joining the origin to the point (x, y) in the complex plane.
In this case, we have:
x = -8 y = 8√3
To find the argument, we can use the following formula:
θ = arctan(y/x)
However, we need to be careful when using this formula, as it may not give the correct answer in all cases.
In this case, we have:
θ = arctan(8√3/-8)
= arctan(√3/-1)
= -π/3
However, we want the argument to lie in the interval [0, 2π). To get this, we can add 2π to the argument:
θ = -π/3 + 2π
= 5π/3
Writing the Answer in Euler's Form
Now that we have found the argument, we can write the answer in Euler's Form:
|z|eiθ = √((-8)^2 + (8√3)2)e(i(5π/3))
= √(64 + 768)e^(i(5π/3))
= √832e^(i(5π/3))
= 4√26e^(i(5π/3))
Therefore, the exact answer to the expression [2(cos(π/3) + isin(π/3))]^4 is 4√26e^(i(5π/3)).
Conclusion
Introduction
DeMoivre's Theorem is a powerful tool in complex analysis that allows us to raise complex numbers to powers. In our previous article, we used DeMoivre's Theorem to calculate the expression [2(cos(π/3) + isin(π/3))]^4 and wrote the exact answer in the form found using Euler's Formula, |z|eiθ. In this article, we will answer some frequently asked questions about DeMoivre's Theorem.
Q: What is DeMoivre's Theorem?
A: DeMoivre's Theorem is a mathematical formula that allows us to raise complex numbers to powers. It states that for any complex number z = r(cosθ + isinθ) and any integer n, we have:
z^n = r^n(cos(nθ) + isin(nθ))
Q: How do I use DeMoivre's Theorem?
A: To use DeMoivre's Theorem, you need to follow these steps:
- Write the complex number in polar form: z = r(cosθ + isinθ)
- Identify the values of r and θ
- Identify the value of n
- Plug these values into the formula: z^n = r^n(cos(nθ) + isin(nθ))
Q: What is the range of DeMoivre's Theorem?
A: DeMoivre's Theorem is valid for any integer n. However, the range of the theorem is limited to the interval [0, 2π) for the argument θ.
Q: Can I use DeMoivre's Theorem for non-integer powers?
A: No, DeMoivre's Theorem is only valid for integer powers. If you need to raise a complex number to a non-integer power, you will need to use a different method, such as the polar form of complex numbers.
Q: How do I find the argument of a complex number?
A: To find the argument of a complex number z = x + iy, you can use the following formula:
θ = arctan(y/x)
However, be careful when using this formula, as it may not give the correct answer in all cases.
Q: Can I use DeMoivre's Theorem to find the roots of a complex number?
A: Yes, DeMoivre's Theorem can be used to find the roots of a complex number. To do this, you need to use the formula:
z^(1/n) = r^(1/n)(cos((θ + 2kπ)/n) + isin((θ + 2kπ)/n))
where k is an integer.
Q: What are some common applications of DeMoivre's Theorem?
A: DeMoivre's Theorem has many applications in mathematics and engineering, including:
- Calculating the roots of complex numbers
- Finding the powers of complex numbers
- Solving trigonometric equations
- Analyzing electrical circuits
Conclusion
In this article, we answered some frequently asked questions about DeMoivre's Theorem. We hope that this article has been helpful in understanding this important mathematical concept.
Additional Resources
For more information on DeMoivre's Theorem, we recommend the following resources:
- "Complex Analysis" by Serge Lang
- "DeMoivre's Theorem" by Wolfram MathWorld
- "Complex Numbers" by Khan Academy
We hope that this article has been helpful in your studies of complex analysis. If you have any further questions, please don't hesitate to ask.